Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests

Abstract

A commutative doubly-idempotent semiring (cdi-semiring) $(S,\vee ,\cdot ,0,1)$ is a semilattice $(S,\vee ,0)$ with $x\vee 0 =x$ and a semilattices $(S,\cdot ,1)$ with identity 1 such that $x0=0$, and $x(y\vee z)=xy \vee xz$ holds for all $x,y,z\in S$. Bounded distributive lattices are cdi-semirings that satisfy $xy=x\wedge y$, and the variety of cdi-semirings covers the variety of bounded distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdi-semirings is generated by a 3-element cdi-semiring. We show that there are seven cdi-semirings with a $\vee$-semilattice of height less than or equal to 2. We construct all cdi-semirings for which their multiplicative semilattice is a chain with $n+1$ elements, and we show that up to isomorphism the number of such algebras is the $n^\text {th}$ Catalan number $C_{n}=\frac{1}{n+1}{2n\atopwithdelims ()n}$. We also show that cdi-semirings with a complete atomic Boolean $\vee$-semilattice on the set of atoms A are determined by singleton-rooted preorder forests on the set A. From these results we obtain efficient algorithms to construct all multiplicatively linear cdi-semirings of size n and all Boolean cdi-semirings of size $2^n$.